It will be shown that the cosmology of the \(Z_{\cal N}\)  axion exhibits several novel aspects.
The cosmological impact of hypothetical parallel “mirror” worlds has been studied at length in the literature (for a review, see e.g. [1]}).
In particular, the constraints on the number of effective relativistic species \(N_{\text{eff}}\)  present in the early Universe imply that the mirror copies of the SM must be less populated –cooler– than the ordinary SM world [2]}.
This requires in turn that the SM has never been in thermal contact with its mirror replica. Fortunately, mechanisms that source this world-asymmetric initial temperatures while preserving the
\(Z_{\cal N}\)  symmetry may arise naturally in the cosmological
evolution [3]}, [4]}, [5]}.
It has also been suggested that DM could be simply constituted by mirror matter,
for which relevant constraints apply given the differences in temperatures.
Note, nevertheless, that in most cases only one replica of the SM was considered,
while large \({\cal N}\)  could significantly modify those analyses.
Furthermore, an axionic nature of DM has not been previously considered in the mirror world setup with a reduced-mass axion.An axion heavier than the QCD one has been contemplated as DM, within a \(Z_2\)  setup which realized linearly the symmetry [6]}.
While gravity and axion-mediated interactions are naturally small enough, the impact of other possible (\(Z_{\cal N}\) -symmetric) interactions on the thermal communication between the SM and its mirror copies will be discussed.
